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Talk:Godgahlah
Is this number comparable to Bowers' goobol = {10,100 (1) 2}? 13:09, September 21, 2012 (UTC) Yes, looks like it. 08:26, September 22, 2012 (UTC) :I'd say so, but only superficially. E# has a different catastrophic rule from BEAF that ultimately grows much slower: {10, 10, 10} (tridecal) = {10, {10, 9, 10}, 9} while E10#10#10 (thridecal?) = E10#(E10#9) FB100Z • talk • 19:36, September 22, 2012 (UTC) ::Uh, never mind. I did stoopid :P FB100Z • talk • 19:38, September 22, 2012 (UTC) ::The details are somewhat tedious and have a few complications involved, but there is good reason to believe that the highest number of consecutive hyper-marks is directly related to the number of entries in array notation. For example, when we restrict ourselves to single hyperions we have: ::E10 = 10^10 = <10,10,1> ::E10#10 > 10^^10 = <10,10,2> ::E10#10#10 > 10^^^10 = <10,10,3> ::... ::E10##n >''' <10,10,n> ::So the number of "proto-hyperions" determines the last entry in a trientrical array. ::Appending additional proto-hyperions to the "deutero-hyperion" ( ## ) results in nested functional powers of E10##n. For example: ::E10##10 '''> <10,10,10> = <10,2,1,2> >''' <10,1,1,2> ::E10##10#2 = E10##(E10##10) '''> E10##(<10,10,10>) >''' <10,10,<10,10,10>> = <10,3,1,2> '''> <10,2,1,2> ::... ::E10##10#n >''' <10,n,1,2> ::... ::E10##10#10 '''> <10,10,1,2> = <10,2,2,2> > <10,1,2,2> ::E10##10#10#2 = E10##10#(E10##10#10) > E10##10#(<10,1,2,2>) '> '<10,<10,1,2,2>,1,2> = <10,2,2,2> ::... ::E10##10#10#n > <10,n,2,2> ::Without being tedious you can begin to see an emerging relation between the functions. ::We can see that: ::E10##10#10#10# ... #10#10#n (w/k 10s after ##) > <10,n,k,2> ::So: ::E10##10##n > <10,10,n-1,2> ::It can be surmised that as we add more deutero-hyperions the 4th entry increases in proportion. In fact: ::E10##10##10## ... ##10 (w/n 10s) ~ <10,10,10,n-1> ::We can see that trito-hyperions ( ### ) would coorspond to 5-entry arrays in short order (which means that numbers like a throogol ''are already beyond chain-arrow expressions). ::So in general it seems that #(k) cooresponds to k+2-entry arrays. Thus a ''godgahlah, E100#(100)100, would be roughly equivalent to 102-entry arrays. Therefore it is almost certainly larger than a goobol. ::An alternative proof is that the addition of every proto-hyperion is the equivalent of going up one rank in the fast-growing hierarchy. This is equivalent to increasing the 3rd entry of any array by 1. We can count how many ranks Extended Hyper-E notation goes up by saying: ::En ~ f_3(n) ::En#n ~ f_4(n) ::En#n#n ~ f_5(n) ::... ::En##n ~ f_w(n) ::En##n#n ~ f_w+1(n) ::En##n#n#n ~ f_w+2(n) ::... ::En##n##n ~ f_2w(n) ::... ::En###n ~ f_w^2(n) ::... ::En###n###n ~ f_2w^2(n) ::... ::En####n ~ f_w^3(n) ::... ::En#(n)n ~ f_w^w(n) ::Since linear arrays are of rank w^w in the fast-growing hierarchy, we can see that Extended Hyper-E is also of the same rank. ::I'll probably work out a complete proof for my website at some point, but I've known for sometime that Extended Hyper-E notation is roughly equivalent to linear array notation and that a godgahlah ''is analogous to the ''goobol. ::''--Sbiis Saibian.'' It is not difficult to show that: {10,2,2} < Googol < {10,3,2} {10,2,1,2} < Gugold < {10,3,1,2} {10,2,1,1,2} < Throogol < {10,3,1,1,2} {10,2,1,1,1,2} < Teroogol < {10,3,1,1,1,2} {10,2,1,1,1,...,1,1,1,2} < Godgahlah < {10,3,1,1,1,...,1,1,1,2} (99 1's, 102-entrical arrays). Ikosarakt1 (talk) 10:57, November 9, 2012 (UTC)